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## Dehn Twist in the sense of Geometric Group Theory and a Graph of Groups

Hello!

Does anyone knows a good book or script about Dehn Twists in the sense of graphs. More precisely: I need to know how a Dehn Twist yields an automorphism of a group or a subgroup of a group. I want to know something about the realtionship between dehn twists and graph of groups.

I know that my request is not nice formulated, but i didn't found any good introduction about this theme in the internet yet. I always get defintions for dehn twists in the sense of surfaces. But i don't know, how to conclude or educe something about groups in the sense of geometric group theory.

Thanks for help.

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In the Handbook of Geometric Topology, Bestvina's article on R-trees has a few pages on this under Applications. There is also the paper Cyclic splittings of finitely presented groups and the canonical JSJ decomposition by Rips and Sela but this is a research paper and I don't think that's what you want.

Informally, a Dehn twist of a group is an element of the automorphism group which functions analogously to a Dehn twist in the mapping class group of a surface. If you have such a surface, you can separate it into smaller surfaces by finding a collection of disjoint simple closed curves and cutting along them. Now, each of the remaining surfaces has a free fundamental group and the fundamental group of the whole space can be computed by van Kampen's theorem. In other words, the original surface is a collection of components and separating annuli which gives you a graph of groups decomposition of the surface group. This is a Z-splitting of the fundamental group.

A Dehn twist, topologically, is performed by taking any loop which intersects more than one of the components above and twisting that curve around the simple closed curve which separates those components. The effect of the Dehn twist in the topological setting is by inner automorphism on the component spaces (you should work out the specific details here).

In the group setting, you have an edge group (which is analogous to the s.c.c. above) by which you conjugate. This gives an automorphism of the group which generalizes the situation described for surfaces above (ie inner on the vertex groups).

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 In your final paragraph, the key point is that you conjugate by an element that centralizes the edge group. – HW Feb 24 2012 at 10:40